PHILOSOPHICAL SOCIETY OF WASHINGTON. tT 



other balls are determined by the conditions that each co-ordi- 

 nate shall be an integer, and that the sum of the co-ordinates 

 shall be an even number. The distance in diameters of any ball 

 centre from the origin is equal to the square root of half the sura 

 of the squares of the co-ordinates. 



In the most general case, the permutation of the co-ordinates 

 gives the factor 6 ; and as the algebraic sign of each co-ordinate 

 may be either positive or negative, the number of variations for 

 each permutation is equal to 8. The product 48 is the number 

 of balls having the same distance and symmetrical arrangement. 

 In particular cases two or all three of the co-ordinates may be 

 numerically equal, or one or more of them may be equal to 0, 

 and the number of such ball centres may be reduced to 6, 8, 12, 

 or 24. Since a number may be the common sum of different 

 sets of squares, there may be more than 48 equidistant balls be- 

 longing to two or more independent symmetrical arrangements. 

 Thus 50 is the common sum of 9, 16, and 25; of 0, 25, and 25; 

 and of 0, 1, and 49; and there are in all 84 ball centres at the 

 distance of 5 diameters from the origin. 



The co-ordinates 0, 1, and 1 give 12 tangent balls, or stars of 

 first magnitude in the imaginary universe; 0, 0, 2 give 6 of 2d 

 magnitude; 1, 1, 2 give 24 of 3d magnitude ; 0, 2, 2 give 12 of 

 4th, magnitude; 0, 1, 3 give 24 of 5th magnitude; 2, 2, 2 give 8 

 •of 6th magnitude; 1, 2, 3 give 48 of 7th magnitude, etc. 



If the origin be regarded as at the centre of a cube whose 

 faces are perpendicular to the co-ordinate axes, the 12 stars of 

 1st magnitude are in the directions of the middle points of the 

 edges, and the 12 of 4th magnitude in right lines beyond those 

 of 1st magnitude; the 6 of 2d magnitude are in the directions of 

 the centres of the faces; and the 8 of 6th magnitude are in the 

 directions of the corners. 



The formula A, b, c may appropriately represent one large co- 

 ordinate and two small ones ; and the corresponding constella- 

 tions consist of octagons around the face-centres, becoming 

 squares when h is numerically equal to c, or when either is equal 

 to 0. The formula A, B, c, denoting two large co-ordinates and 

 one small one, corresponds to rectangles about the middle points 

 of the edges, becoming pairs when A is numerically equal to B, 

 or when c is equal to 0. The formula A, B, G, denoting co-ordi- 

 nates nearly equal, corresponds to hexagons about the corners, 

 which are regular when A, B, and G are numerically in arithme- 

 tical progression, and become triangles when two of the co-ordi- 

 nates are numerically equal. 



Mr. Ball's communication On the Boundary Line between 

 Alaska and British America, having been made on the spur of 

 the moment to fill an unoccupied hour of the evening, he desired 

 no further mention than the entrance of the title upon the 

 minutes of the meeting. 



The Society then adjourned. 



